For the dauphinoise, preheat the oven to 175C/350F/Gas 3.
Place the potatoes into a bowl of cold water and swirl around to remove any excess starch. Drain well and pat dry.
Butter an ovenproof dish with about a teaspoon of the butter and place the potatoes in the dish in layers, overlapping a little, until the dish is just full - season with salt, freshly ground black pepper and nutmeg between each layer.
Whisk the cream and milk in a bowl until well combined. Season with salt, freshly ground black pepper and nutmeg and pour the cream and milk over the potatoes. Dot with the remaining butter, then cover tightly with aluminium foil.
Bake in the oven for one hour, or until the potatoes are just tender.
After an hour, carefully remove the foil and return to the oven for a further 30 minutes, or until golden-brown on top. Remove from the oven and set aside to cool. Stamp out four 7.5cm/3in rounds. (The leftovers can be eaten at a later date).
For the pithivier, preheat the oven to 200C/400F/Gas 6.
Heat a frying pan until hot, add 25g/1oz of the butter and the wild mushrooms and fry for 2-3 minutes, or until just softened. Stir in the tarragon and season with salt and freshly ground black pepper. Remove the pan from the heat and set aside to cool slightly.
Roll the puff pastry out on a floured surface to a 5mm/⅛in thickness. Cut out four 10cm/4in circles and four 15cm/6in circles.
Place the small circles onto a baking tray and top with the rounds of dauphinoise. Add a spoonful of mushrooms and a slice of tallegio cheese. Brush the edge of the pastry with the beaten egg.
Carefully lay the large circle of pastry over the top of each pile, press down lightly over the mound and around the edge to seal. Mark small semi-circles on the pastry with a table knife to create a shell pattern. Brush with the remaining egg and sprinkle with a little salt.
Bake in the oven for 20-30 minutes, or until hot through and the pastry is crisp and golden-brown.
Heat a frying pan until hot, add the remaining butter, spring greens and water and fry for 2-3 minutes, or until just wilted.
Stir in the remaining mushrooms.
To serve, spoon the spring greens onto serving plates and serve the pithivier alongside.